olpea,
I've got some notes around here somewhere on 'gender', parthenocarpy, and the like, that I'll try to dig up and post.
Staminate flowers on persimmon look not unlike the female flowers, but occur in clusters of three rather than as singles.

Found this one, from the late Lon Rombough:

"More than one report suggests that persimmon flowers may be stimulated to set seedless fruit by pollen of other species, particularly chestnuts (but this has not been proven). It's possible yours is setting seedless fruit this way. The fact that there is some seed shows that pollen from a male persimmon is reaching it someplace. But since many persimmons, particularly ones related to Early Golden, are able to produce limbs of male flowers, you may have a male in the area withough knowing it.
Climate also has an effect. Here in Oregon "Meader" will set seedless fruit with no male around, but in many areas it must have a male to set any fruit.
It does produce limbs of male flowers, though, so a fruitless tree may suddenly start bearing after a few years, when a male limb arises. In fact, it seems that it's a survival tactic of isolated American persimmon trees to be able to produce flowers of the opposite sex in some years. Males are known that produce perfect flowers some years, and some males will produce small, seedless fruit almost every year."

On that point, I have one local native tree that produces 80% small (1") seedless fruits, and about 20% larger 1.5" fruits that usually only contain a single plump seed. I can never think to stop and look when it's blooming to figure out if it's a male or what - though I've sent scionwood to Jerry Lehman, and he says all flowers on his resulting trees were female. It's just an oddity, I guess.

Thanks Lucky, good information.

I'll make a mental note to check the tree next year for any limbs which have clusters of three (male) flowers.

I haven't cut any fruit open to check for seeds. The fruit is small, but I understand Morris Burton fruit are unusually small anyway.